I recently asked (and then answered) this question:

In a separable metric space there is no strictly decreasing sequence of closed sets $(X_\alpha)_{\alpha<\omega_1}$, where $X_\beta\supsetneq X_\alpha$ iff $\beta<\alpha$.

This sounds a lot like the Descending Chain Condition in Algebra. I assume it comes up frequently in topology (maybe not?). In that case, is there a name for it?